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arXiv:math-ph/0612055v1 17 Dec 2006
THE LANGEVIN EQUATION
FOR A QUANTUM HEAT BATH
St´ ephane ATTAL
1& Alain JOYE
2
1Institut C. Jordan
Universit´ e C. Bernard, Lyon 1
21, av Claude Bernard
69622 Villeurbanne Cedex
France
2Institut Fourier
Universit´ e de Grenoble 1
100, rue des Maths, BP 74
38402 St Martin d’Heres
France
Abstract
We compute the quantum Langevin equation (or quantum stochastic differential equation) repre-
senting the action of a quantum heat bath at thermal equilibrium on a simple quantum system. These
equations are obtained by taking the continuous limit of the Hamiltonian description for repeated
quantum interactions with a sequence of photons at a given density matrix state. In particular we spe-
cialise these equations to the case of thermal equilibrium states. In the process, new quantum noises
are appearing: thermal quantum noises. We discuss the mathematical properties of these thermal
quantum noises. We compute the Lindblad generator associated with the action of the heat bath on
the small system. We exhibit the typical Lindblad generator that provides thermalization of a given
quantum system.
I. Introduction
The aim of Quantum Open System theory (in mathematics as well as in
physics) is to study the interaction of simple quantum systems interacting with
very large ones (with infinite degrees of freedom). In general the properties that
one is seeking are to exhibit the dissipation of the small system in favor of the
large one, to identify when this interaction gives rise to a return to equilibrium or
a thermalization of the small system.
There are in general two ways of studying those system, which usually repre-
sent distinct groups of researchers (in mathematics as well as in physics).
The first approach is Hamiltonian. The complete quantum system formed by
the small system and the reservoir is studied through a Hamiltonian describing
the free evolution of each component and the interaction part. The associated
unitary group gives rise to a group of *-endomorphisms of a certain von Neumann
algebra of observables. Together with a state for the whole system, this constitutes
a quantum dynamical system. The aim is then to study the ergodic properties of
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that quantum dynamical system. This can be performed via the spectral study of
a particular generator of the dynamical system: the standard Liouvillian. This is
the only generator of the quantum dynamical system which stabilizes the self-dual
cone of the associated Tomita-Takesaki modular theory. It has the property to
encode in its spectrum the ergodic behavior of the quantum dynamical system.
Very satisfactory recent results in that direction were obtained by Jaksic and Pillet
([JP1], [JP2] and [JP3]) who rigorously proved the return to equilibrium for Pauli-
Fierz systems, using these techniques.
The second approach is Markovian. In this approach one gives up the idea of
modelizing the reservoir and concentrates on the effective dynamics of the small
system. This evolution is supposed to be described by a semigroup of completely
positive maps. These semigroups are well-known and, under some conditions,
admit a generator which is of Lindblad form:
L(X) = i[H,X] +1
2
i
The first order part of L represents the usual quantum dynamic part, while the
second order part of L carries the dissipation. This form has to be compared with
the general form, in classical Markov process theory, of a Feller diffusion generator:
a first order differential part which carries the classical dynamics and a second
order differential part which represents the diffusion. For classical diffusion, such
a semigroup can be realized as resulting of a stochastic differential equation. That
is, a perturbation of an ordinary differential equation by classical noise terms such
as a Brownian motion usually. In our quantum context, one can add to the small
system an adequate Fock space which carries quantum noises and show that the
effective dynamics we have started with is resulting of a unitary evolution on the
coupled system, driven by a quantum Langevin equation. That is, a perturbation
of a Schr¨ odinger-type equation by quantum noise terms.
?
(2L∗
iXLi− L∗
iLiX − XL∗
iLi).
Whatever the approach is, the study of the action of quantum thermal baths is
of major importance and has many applications. In the Hamiltonian approach, the
model for such a bath is very well-known since Araki-Woods’ work ([A-W]). But
in the Markovian context, it was not so clear what the correct quantum Langevin
equation should be to account for the action of a thermal bath. Some equations
have been proposed, in particular by Lindsay and Maassen ([L-M]). But no true
physical justification of them has ever been given. Besides, it is not so clear what
a “correct” equation should mean?
A recent work of Attal and Pautrat ([AP1]) is a good candidate to answer
that problem. Indeed, consider the setup of a quantum system (such as an atom)
having repeated interactions, for a short duration τ, with elements of a sequence
of identical quantum systems (such as a sequence of photons). The Hamiltonian
evolution of such a dynamics can be easily described. It is shown in [AP1] that in
the continuous limit (τ → 0), this Hamiltonian evolution spontaneously converges
to a quantum Langevin equation. The coefficient of the equation being easily
computable in terms of the original Hamiltonian. This work has two interesting
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consequences:
– It justifies the Langevin-type equations for they are obtained without any
probabilistic assumption, directly from a Hamiltonian evolution;
– It is an effective theorem in the sense that, starting with a naive model for
a quantum field (a sequence of photons interacting one after the other with the
small system), one obtains explicit quantum Langevin equations which meet all
the usual models of the litterature.
It seems thus natural to apply this approach in order to derive the correct
quantum Langevin equations for a quantum heat bath. This is the aim of this
article.
We consider a simple quantum system in interaction with a toy model for
a heat bath. The toy model consists in a chain of independent photons, each of
which in the thermal Gibbs state at inverse temperature β, which are interacting
one after the other with the small system. Passing to the continuous interaction
limit, one should obtain the correct Langevin equation.
One difficulty here is that in [AP1], the state of each photon needed to be
a pure state (this choice is crucial in their construction). This is clearly not the
case for a Gibbs state. We solve this problem by taking the G.N.S. (or cyclic)
representation associated to that state. If the state space of one (simplified) photon
was taken to be n-dimensional, then taking the G.N.S. representation brings us
into a n2-dimensional space. This may seem far too big and give the impression
we will need too many quantum noises in our model. But we show that, in all
cases, only 2n chanels of noise resist to the passage to the limit and that they can
be naturally coupled two by two to give rise to n “thermal quantum noises”. The
Langevin equation then remains driven by n noises (which was to be expected!)
and the noises are shown to be actually Araki-Woods representations of the usual
quantum noises. Furthermore, the Langevin equation we obtain is very similar to
the model given in [L-M].
Altogether this confirms we have identified the correct Langevin equation
modelizing the action of a quantum heat bath.
An important point to notice is that our construction does not actually use
the fact that the state is a Gibbs-like state, it is valid for any density matrix.
This article is organized as follows. In section II we present the toy model for
the bath and the Hamiltonian description of the repeated interaction procedure.
In section III we present the Fock space, its quantum noises, its approximation by
the toy model and the main result of [AP1]. In section IV we detail the G.N.S.
representation of the bath and compute the unitary operator, associated with the
total Hamiltonian, in that representation. In section V, applying the continuous
limit procedure we derive the limit quantum langevin equation. In the process, we
identify particular quantum noises that are naturally appearing and baptize them
“thermal quantum noises”, in the case of a heat bath. The properties of those
thermal quantum noises are studied in section VI; in particular we justify their
name. In section VII, tracing out the noise, we compute the Lindblad generator of
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the induced semigroup on the small system. In section VII, being given any finite
dimensional quantum system with its Hamiltonian, we show how to construct a
Lindblad generator, representing some interaction with a heat bath, such that the
quantum system thermalizes.
II. The toy model
We describe here the physical model of repeated interactions with the bath
toy model.
The quantum system (we shall often call “small system”) to be put in contact
with the bath is represented by a separable Hilbert space HS, as state space, and
a self-adjoint operator HS, as Hamiltonian.
The toy model for the heat bath is the chain
?
of copies of CN+1, where N ≥ 1 is a fixed integer. Each copy of CN+1represents
the (simplified) state space of a photon. By this countable tensor product we
mean the following. We consider a fixed orthonormal basis {e0,e1,...,eN} of
CN+1, corresponding to the eigenstates of the photon (e0being the ground state);
the countable tensor product is taken with respect to the ground state e0. Together
with this structure we consider the associated basic matrices ai
acting on CN+1by
ai
jek= δikej
and their natural ampliations to ⊗k∈I N∗CN+1given by
ai
j(k) =
I on the other copies.
The Hamiltonian of one photon is the operator
k∈I N∗
CN+1
j, i,j = 1,...,N,
?
ai
j
on the k-th copy of CN+1
HR=
N
?
i=0
γia0
iai
0,
where the γi’s are real numbers. Here notice two points.
We have assumed the Hamiltonian HR to be diagonal in the chosen basis.
This is of course not actually a true restriction, for one can always choose such
a basis.Note that HRdescribe the total energy of a single photon, not the whole
field of photon. For this we differ from the model studied in [AJ1].
The second point is that γ0is the ground state eigenvalue, it should then be
smaller than the other γi. One usually assumes that it is equal to 0, but this is not
actually necessary in our case, we thus do not specify its value. The only hypothesis
we shall make here is that γ0< γi, for all i = 1,...N. This hypothesis means
that the ground eigenspace is simple, it is not actually a necessary assumption, it
only simplifies our discussion. At the end of section V we discuss what changes if
we leave out this hypothesis.
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Finally, notice that the other eigenvalues γineed not be simple in our discus-
sion.
When the system and a photon are interacting, we consider the state space
HS⊗ CN+1together with the interaction hamiltonian
N
?
where the Vi’s are bounded operators on HS. This is a usual dipole-type interaction
Hamiltonian. The total Hamiltonian for the small system and one photon is thus
HI=
i=1
?Vi⊗ a0
i+ V∗
i⊗ ai
0
?,
H = HS⊗ I + I ⊗ HR+
N
?
i=1
?Vi⊗ a0
i+ Vi⊗ ai
0
?.
Finally, the state of each photon is fixed to be given by a density matrix ρ which
is a function of HR. We have in mind the usual thermal Gibbs state at inverse
temperature β :
ρβ=1
Ze−βHR,
where Z = tr(e−βHR), but our construction applies to more general states ρ.
Note that ρβis also diagonal in our orthonormal basis. Its diagonal elements
are denoted by {β0,β1,...,βn}.
We shall now describe the repeated interactions of the system HS with the
chain of photons. The system HS is first in contact with the first photon only
and they interact together according to the above Hamiltonian H. This lasts for
a time length τ. The system HSthen stops interacting with the first photon and
starts interacting with the second photon only. This second interaction is directed
by the same Hamiltonian H on the corresponding spaces and it lasts for the same
duration τ, and so on... This is mathematically described as follows.
On the space HS⊗ CN+1, consider the unitary operator representing the
coupled evolution during the time interval [0,τ]:
U = e−iτH.
This single interaction is therefore described in the Schr¨ odinger picture by
ρ ?→ U ρU∗
and in the Heisenberg picture by
X ?→ U∗XU.
After this first interaction, we repeat it but coupling the same HSwith a new copy
of CN+1. This means that this new copy was kept isolated until then; similarly
the previously considered copy of CN+1will remain isolated for the rest of the
experience.
The sequence of interactions can be described in the following way: the state
space for the whole system is
?
HS⊗
I N∗
CN+1.
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Consider the unitary operator Uk which acts as U on the tensor product of HS
and the k-th copy of CN+1, and which acts as the identity on all the other copies
of CN+1.
The effect of the k-th interaction in the Schr¨ odinger picture is
ρ ?→ UkρU∗
for every density matrix ρ on HS⊗∗
interactions is
ρ ?→ VkρV∗
where Vk= UkUk−1...U1.
k,
I NCn+1. In particular the effect of the k first
k
Such a Hamiltonian description of the repeated interaction procedure has no
chance to give any non-trivial limit in the continuous limit (τ → 0) without asking
a certain renormalization of the interaction. This renormalization can be thought
of as making the Hamiltonian depend on τ, or can be also seen as renormalizing
the field operators a0
0of the photons. As is shown is [AP1] (see the detailed
discussion in section III), for our repeated interaction model to give rise to a
Langevin equation in the limit, we need the interaction part of the Hamiltonian
to be affected by a weight 1/√τ. Hence, from now on, the total Hamiltonians we
shall consider on HS⊗ CN+1are
1
√τ
j,ai
H = HS⊗ I + I ⊗ HR+
N
?
i=1
?Vi⊗ a0
i+ V∗
i⊗ ai
0
?. (1)
In [AJ1], one can find a discussion about this time renormalization and its inter-
pretation in terms of weak coupling limit for repeated quantum interactions.
III. The continuous limit setup
We present here all the elements of the continuous limit result: the structure
of the corresponding Fock space, the quantum noises, the approximation of the
Fock space by the photon chain and [AP1]’s main theorem.
III.1 The continuous tensor product structure
First, as a guide to intuition, let us make more explicit the structure of the
photon chain. We let TΦ denote the tensor product ⊗I N∗CN+1with respect to
the stabilizing sequence e0. This simply means that an orthonormal basis of TΦ
is given by the family
{eσ;σ ∈ PI N∗,N}
where
– the set PI N,N is the set of finite subsets
{(n1,i1),...,(nk,ik)}
of I N∗× {1,...,N} such that the ni’s are mutually different;
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– eσdenotes the vector
Ω ⊗ ... ⊗ Ω ⊗ ei1⊗ Ω ⊗ ... ⊗ Ω ⊗ ei2⊗ ...
where ei1appears in n1-th copy of H, where ei2appears in n2-th copy of H... Here
Ω plays the same role as e0in the toy model.
This is for a vector basis on TΦ. From the point of view of operators, we
denote by ai
copy number k as ai
jand the identity elsewhere. That is, in terms of the basis eσ,
ai
j(k)eσ= 1 l(k,i)∈σe(σ\(k,i))∪(k,j)
if neither i nor j is zero, and
ai
0(k)eσ= 1 l(k,i)∈σeσ\(k,i),
a0
j(k)eσ= 1 l(k,0)∈σeσ∪(k,j),
a0
0(k)eσ= 1 l(k,0)∈σeσ,
where (k,0) ∈ σ actually means “for any i in {1,...,N}, (k,i) ?∈ σ”.
We now describe the structure of the continuous version of the chain of pho-
tons. The structure we are going to present here is rather original and not much
expanded in the literature. It is very different from the usual presentation of quan-
tum stochastic calculus ([H-P]), but it actually constitutes a very natural language
for our purpose: approximation of the atom field by atom chains. This approach
is taken from [At1]. We first start with a heuristic discussion.
By a continuous version of the atom chain TΦ we mean a Hilbert space with
a structure which makes it the space
?
We have to give a meaning to the above notation. This could be achieved by in-
voquing the framework of continous tensor products of Hilbert spaces (see [Gui]),
but we prefer to give a self-contained presentation which fits better with our ap-
proximation procedure.
Let us make out an idea of what it should look like by mimicking, in a con-
tinuous time version, what we have described in TΦ.
The countable orthonormal basis eσ,σ ∈ PI N∗,N is replaced by a continuous
orthonormal basis dχσ, σ ∈ PI R+,N, where PI R+,Nis the set of finite subsets of
I R+× {1,...,N}. With the same idea as for TΦ, this means that each copy of
CN+1is equipped with an orthonormal basis {Ω,dχ1
parameter attached to the copy we are looking at).
Recall the representation of an element f of TΦ:
?
||f||2=
σ∈PI N∗,N
j(k) the natural ampliation of the operator ai
jto TΦ which acts on the
Φ =
I R+
CN+1.
t,...,dχN
t} (where t is the
f =
σ∈PI N∗,N
?
f(σ)eσ,
|f(σ)|2,
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it is replaced by an integral version of it in Φ:
f =
?
?
PI R+,N
f(σ)dχσ,
||f||2=
PI R+,N
|f|2dσ.
This last integral needs to be explained: the measure dσ is a “Lebesgue measure”
on PI R+,N, as will be explained later.
From now on, the notation P will denote, depending on the context, the set
PI N∗,N or PI R+,N.
A good basis of operators acting on Φ can be obtained by mimicking the
operators ai
{0,1,...,N}, acting on the “t-th” copy of CN+1by:
da0
0(t)dχσ= dχσdt1 lt?∈σ
da0
i(t)dχσ= dχσ∪{(t,i)}1 lt?∈σ
dai
0(t)dχσ= dχσ\{(t,i)}dt1 l(t,i)∈σ
dai
j(t)dχσ= dχ(σ\{(t,i)})∪{(t,j)}1 l(t,i)∈σ
for all i,j ∈ {1,...,N}. We shall now describe a rigourous setup for the above
heuristic discussion.
j(k) of TΦ. We have here a set of infinitesimal operators dai
j(t), i,j ∈
We recall the structure of the bosonic Fock space Φ and its basic structure
(cf [At1] for more details and [At2] for a complete study of the theory and its
connections with classical stochastic processes).
Let Φ = Γs(L2(I R+,CN)) be the symmetric (or bosonic) Fock space over the
space L2(I R+,CN). We shall give here a very efficient presentation of that space,
the so-called Guichardet interpretation of the Fock space.
Let P (= PI R+,N) be the set of finite subsets {(s1,i1),...,(sn,in)} of I R+×
{1,...,N} such that the siare two by two different. Then P = ∪kPkwhere Pk
is the subset of P made of k-elements subsets of I R+× {1,...,N}. By ordering
the I R+-part of the elements of σ ∈ Pk, the set Pk can be identified with the
increasing simplex Σk = {0 < t1< ··· < tk} × {1,...,N} of I Rk× {1,...,N}.
Thus Pkinherits a measured space structure from the Lebesgue measure on I Rk
times the counting measure on {1,...,N}. This also gives a measure structure on
P if we specify that on P0= {∅} we put the measure δ∅. Elements of P are often
denoted by σ, the measure on P is denoted by dσ. The σ-field obtained this way
on P is denoted by F.
We identify any element σ ∈ P with a family {σi, i ∈ {1,...,N}} of (two by
two disjoint) subsets of I R+where
σi= {s ∈ I R+;(s,i) ∈ σ}.
The Fock space Φ is the space L2(P,F,dσ). An element f of Φ is thus a
measurable function f : P → C such that
||f||2=
P
?
|f(σ)|2dσ < ∞.
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Finally, we put Ω to be the vacuum vector of Φ, that is, Ω(σ) = δ∅(σ).
One can define, in the same way, P[a,b]and Φ[a,b]by replacing I R+with
[a,b] ⊂ I R+. There is a natural isomorphism between Φ[0,t]⊗Φ[t,+∞[and Φ given
by h⊗g ?→ f where f(σ) = h(σ ∩[0,t])g(σ∩[t,+∞[). This is, with our notations,
the usual exponential property of Fock spaces. Note that in the sequel we identify
Φ[a,b]with a subspace of Φ, the subspace
{f ∈ Φ;f(σ) = 0 unless σ ⊂ [a,b]}.
We now define a particular family of curves in Φ, which is going to be of great
importance here. Define χi
t∈Φ by
χi
t(σ) =
0
Then notice that for all t ∈ I R+we have that χi
have much more than that:
χi
s∈ Φ[s,t]for all s ≤ t.
This last property can be checked immediately from the definitions, and it is
going to be of great importance in our construction. Also notice that χi
are orthogonal elements of Φ as soon as i ?= j. One can show that, apart from
trivialities, the curves (χi
t)t≥0are the only ones to share these properties.
These properties allow to define the so-called Ito integral on Φ. Indeed, let
g = {(gi
and {1,...,N}, such that
i) t ?→ ?gi
ii) gi
t∈Φ[0,t]for all t,
iii)?N
N
?
to be the limit in Φ of
N
?
where S = {tj, j∈I N} is a partition of I R+which is understood to be refining
and to have its diameter tending to 0, and (? gi
Note that by assumption we always have that ? gi
Also note that, as an example, one can take
?
?1 l[0,t](s) if σ = {(s,i)}
otherwise.
tbelongs to Φ[0,t]. We actually
t− χi
tand χj
s
t)t≥0, i ∈ {1,...,N}} be families of elements of Φ indexed by both I R+
t? is measurable, for all i,
?∞
i=1
0?gi
t?2dt < ∞,
then one says that g is Ito integrable and we define its Ito integral
i=1
?∞
0
gi
tdχi
t
i=1
?
j∈I N
? gi
tj⊗
?
χi
tj+1− χi
tj
?
(2)
·)iis an Ito integrable family in Φ,
tis a step process, and which converges to (gi
such that for each i, t ?→ ? gi
χi
tjbelongs to Φ[tj,tj+1], hence the tensor product symbol in (2).
·)i in
L2(I R+× P).
tjbelongs to Φ[0,tj]and χi
tj+1−
? gi
t=
tj∈S
1
tj+1− tj
?tj+1
tj
Ptjgi
sds1 l[tj,tj+1[(t)
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if t ∈ [tj,tj+1], where Ptdenotes the orthogonal projection onto Φ[0,t].
One then obtains the following properties ([At1], Proposition 1.4), where ∨σ
means max{s ∈ I R+;(s,k) ∈ σ for some k} and where σ− denotes the set σ\(∨σ,i)
if (∨σ,i) ∈ σ.
Theorem 1.–The Ito integral I(g) =?
I(g)(σ) =
gi
∨σ(σ−)
It satisfies the Ito isometry formula:
???
In particular, consider a family f = (fi)N
L2(I R+× {1,...,N}), then the family (fi(t)Ω), t ∈ I R+, i = 1,...,N, is clearly
Ito integrable. Computing its Ito integral we find that
?∞
is the element of the first particle space of the Fock space Φ associated with the
function f, that is,
?fi(s)
Let f∈L2(Pn), one can easily define the iterated Ito integral on Φ:
In(f) =
Pn
by iterating the definition of the Ito integral:
?
We obtain this way an element of Φ which is actually the representant of f in the
n-particle subspace of Φ, that is
?fi1,...,in(t1,...,tn)
Finally, for any f ∈ P we put
?
to denote the series of iterated Ito integrals
?∞
i
?∞
0gi
tdχi
t, of an Ito integrable family
g = (gi
·)N
i=1, is the element of Φ given by
?0 if σ = ∅
otherwise.
||I(g)||2=
N
?
i=1
?∞
0
gi
tdχi
t
???
2
=
N
?
i=1
?∞
0
????gi
t
????2dt . (3)
i=1which belongs to L2(P1) =
I(f) =
N
?
i=1
0
fi(t)Ωdχi
t
I(f)(σ) =
if σ = {(s,i)}
otherwise.0
?
f(σ)dχσ
In(f) =
i1,...,in∈{1,...,N}
?∞
0
?tn
0
...
?t2
0
fi1,...,in(t1,...,tn)Ω dχi1
t1... dχin
tn.
[In(f)](σ) =
if σ = {(t1,i1) ∪ ... ∪ (tn,in)}
otherwise.0
P
f(σ)dχσ
f(∅)Ω +
∞
?
n=1
N
?
i1,...,in=1
0
?tn
0
...
?t2
0
fi1,...,in(t1,...,tn)Ωdχi1
t1... dχin
tn.
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We then have the following representation ([At1], Theorem 1.7).
Theorem 2. [Fock space chaotic representation property]–Any element f of Φ
admits a Fock space chaotic representation
?
satisfying the isometry formula
?
This representation is unique.
f =
P
f(σ) dχσ
(4)
?f?2=
P
|f(σ)|2dσ. (5)
The above theorem is the exact expression of the heuristics we wanted in order
to describe the space
Φ =
?
I R+
H.
Indeed, we have, for each t ∈ I R+, a family of elementary orthonormal elements
{Ω,dχ1
basis of Φ (formula (4)) and, even more, form an orthonormal continuous basis
(formula (5)).
t,...,dχN
t} (a basis of H) whose (tensor) products dχσform a continuous
III.2 The quantum noises
The space Φ we have constructed is the natural space for defining quantum
noises. These quantum noises are the natural, continuous-time, extensions of the
basis operators ai
j(n) we met in the atom chain TΦ.
As indicated in the heuristic discussion above, we shall deal with a family of
infinitesimal operators dai
j(t) on Φ which act on the continuous basis dχσin the
same way as their discrete-time counterparts ai
version of the above heuristic infinitesimal formulas easily gives an exact formula
for the action of the operators ai
j(t) on Φ:
?
s≤t
?t
[ai
j(t)f](σ) =
s∈σi
s≤t
[a0
0(t)f](σ) = tf(σ)
for i,j ?= 0.
All these operators, except a0
0(t), are unbounded, but note that a good com-
mon domain to all of them is
?
11
j(n) act on the eσ. The integrated
[a0
i(t)f](σ) =
s∈σi
f(σ \ (s,i)),
[ai
0(t)f](σ) =
0
f(σ ∪ (s,i)) ds,
?
f ((σ \ (s,i)) ∪ (s,j))
D =f∈Φ ;
?
P
|σ| |f(σ)|2dσ < ∞
?
.
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This family of operators is characteristic and universal in a sense which is close
to the one of the curves χi
t. Indeed, one can easily check that in the decomposition
of Φ ≃ Φ[0,s]⊗ Φ[s,t]⊗ Φ[t,+∞[, the operators ai
I ⊗ (ai
This property is fundamental for the definition of the quantum stochastic integrals
and, in the same way as for (χi
·), these operator families are the only ones to share
that property (cf [Coq]).
This property allows to consider Riemann sums:
?
where S = {0 = t0< t1< ... < tk< ...} is a partition of I R+, where (Ht)t≥0is a
family of operators on Φ such that
– each Ht is an operator of the form Ht⊗ I in the tensor product space
Φ = Φ[0,t]⊗ Φ[t,+∞[(we say that Htis a t-adapted operator and that (Ht)t≥0is
an adapted process of operators),
– (Ht)t≥0is a step process, that is, it is constant on intervals:
Ht=Htk1 l[tk,tk+1](t).
j(t) − ai
j(s) are all of the form
j(t) − ai
j(s))|Φ[s,t]⊗ I.
k
Htk
?ai
j(tk+1) − ai
j(tk)?
(6)
?
k
In particular, the operator product Htk
product of operators
?ai
j(tk+1) − ai
j(tk)?.
j(tk)?
is actually a tensor
Htk⊗?ai
j(tk+1) − ai
Thus this product is commutative and does not impose any new domain constraint
on the operators apart from the ones attached to the operators Htand ai
ai
j(tk) themselves. The resulting operator associated to the Riemann sum (6) is
denoted by
?∞
One can compute the action of T on a “good” vector f of its domain and obtain
explicit formulas which are not worth developping here (cf [At1] for more details).
For general operator processes (Ht)t≥0(still adapted but not a step process any-
more) and for a general f, these explicit formulas can be extended and they are
kept as a definition for the domain and for the action of the operator
?∞
The maximal domain and the explicit action of the above operator can be de-
scribed but also are not worth developing here (cf [A-L]). The main point with
these quantum stochastic integrals is that, when composed, they satisfy a Ito-type
integration by part formula. This formula can be summarized as follows, without
taking care at all of domain constraints. Let
?∞
j(tk+1)−
T =
0
Hsdai
j(s).
T =
0
Hsdai
j(s).
T =
0
Hsdai
j(s), S =
?∞
0
Ksdak
l(s).
12
Page 13
For every t ∈ I R+put
Tt=
?∞
?∞
?δil
0
Hs1 l[0,t](s)dai
j(s)
and the same for St. We then have
?∞
where
TS =
0
HsSsdai
j(s) +
0
TsKsdak
l(s) +
?∞
0
HsKs?δildak
j(s), (7)
?δil=
if (i,l) ?= (0,0)
if (i,l) = (0,0).0
The last term appearing in this Ito-type formula is often summarized by saying
that the quantum noises satisfy the formal formula:
dai
j(s)dak
l(s) =?δildak
j(s).
III.3 Embedding and approximation by the Toy Fock space
We now describe the way the chain and its basic operators can be realized as a
subspace of the Fock space and a projection of the quantum noises. The subspace
associated with the atom chain is attached to the choice of some partition of I R+
in such a way that the expected properties are satisfied:
– the associated subspaces increase when the partition refines and they con-
stitute an approximation of Φ when the diameter of the partition goes to 0,
– the associated basic operators are restrictions of the others when the parti-
tion increases and they constitute an approximation of the quantum noises when
the diameter of the partition goes to 0.
Let
S = {0 = t0 < t1 < ··· < tn < ···} be a partition of
δ(S) = supi|ti+1− ti| be the diameter of S. For S fixed, define Φn= Φ[tn−1,tn],
n∈I N∗. We clearly have that Φ is naturally isomorphic to the countable tensor
product ⊗n∈I N∗Φn(which is again understood to be defined with respect to the
stabilizing sequence (Ω)n∈I N).
For all n∈I N∗, define for i,j ∈ {1,...,N}
χi
tn−1
√tn− tn−1
ai
√tn− tn−1
ai
I R+and
ei(n) =
tn− χi
∈ Φn,
0(n) =ai
j(n) = P1]◦?ai
a0
0(tn) − ai
0(tn−1)
◦ P1],
j(tn−1)?◦ P1],
j(tn) − ai
i(tn) − a0
√tn− tn−1
i(n) = P1]◦a0
a0
0(n) = P0],
i(tn−1)
,
where for i = 0,1 and Pi]is the orthogonal projection onto L2(Pi). The above
definitions are understood to be valid on Φn only, the corresponding operator
acting as the identity operator I on the others Φm’s.
13
Page 14
For every σ ∈ P = PI N∗,N, define eσfrom the ei(n)’s in the same way as for
TΦ:
eσ= Ω ⊗ ... ⊗ Ω ⊗ ei1(n1) ⊗ Ω ⊗ ... ⊗ Ω ⊗ ei2(n2) ⊗ ...
in ⊗n∈I N∗Hn. Define TΦ(S) to be the space of f∈Φ which are of the form
f =
?
σ∈P|f(σ)|2< ∞ is automatically satisfied).
σ∈P
f(σ)eσ
(note that the condition ?f?2=?
The space TΦ(S) is a closed subspace of Φ. We denote by PS the operator of
orthogonal projection from Φ onto TΦ(S).
The main point is that the above operators ai
way as the the basic operators of TΦ (cf [AP1], Proposition 8).
The space TΦ(S) can be clearly and naturally identified to the spin chain TΦ.
j(n) act on TΦ(S) in the same
Proposition 3.–We have, for all i,j = 1,...,N
?ai
?ai
?a0
?a0
0(n)ej(n) = δijΩ
ai
0Ω = 0
j(n)ek(n) = δikej(n)
ai
jΩ = 0
i(n)ej(n) = 0
a0
i(n)Ω = ei(n)
0(n)ek(n) = 0
a0
0Ω = Ω.
Thus the action of the operators ai
action of the corresponding operators on the spin chain of section II; the operators
ai
j(n) act on TΦ(S) exactly in the same way as the corresponding operators do
on TΦ. We have completely embedded the toy Fock space structure into the Fock
space.
jon the ei(n) is exactly the same as the
We are now going to see that the Fock space Φ and its basic operators ai
i,j ∈ {0,1,...,n} can be approached by the toy Fock spaces TΦ(S) and their
basic operators ai
j(n). We are given a sequence (Sn)n∈I N of partitions which are
getting finer and finer and whose diameter δ(Sn) tends to 0 when n tends to +∞.
Let TΦ(n) = TΦ(Sn) and Pn= PSn, for all n∈I N. We then have the following
convergence result (see [AP1], Theorem 10), where the reader needs to recall the
domain D introduced in section III.2.
j(t),
Theorem 4.–
i) The orthogonal projectors Pn converge strongly to the identity operator I
on Φ. That is, any f ∈ Φ can be approached in Φ by a sequence (fn)n∈I Nsuch
that fn∈ TΦ(n) for all n ∈ I N.
14
Page 15
ii) If Sn= {0 = tn
1,...,n the operators
0< tn
1< ··· < tn
?
?
?
and
k< ···}, then for all t∈I R+, all i,j =
k;tn
k≤t
ai
j(k),
k;tn
k≤t
?
?
(tn
tn
k− tn
k−1ai
0(k),
k;tn
?
k≤t
tn
k− tn
k−1a0
i(k)
k;tn
k≤t
0(t), a0
k− tn
k−1)a0
0(k)
converge strongly on D to ai
We have fulfilled our duties: not only the space TΦ(S) recreates TΦ and its
basic operators as a subspace of Φ and a projection of its quantum noises, but,
when δ(S) tends to 0, this realisation constitutes an approximation of the space
Φ and of its quantum noises.
j(t), ai
i(t) and a0
0(t) respectively.
III.4 Quantum Langevin equations
In this article what we call quantum Langevin equation is actually a restricted
version of what is usually understood in the physical literature (cf [G-Z]); by this we
mean that we study here the so-called quantum stochastic differential equations as
defined by Hudson and Parthasarathy and heavily studied by further authors ([H-
P], [Fag]). This type of quantum noise perturbation of the Schr¨ odinger equation
is exactly the type of equation which we will get as the continuous limit of our
Hamiltonian description of repeated quantum interactions.
Quantum stochastic differential equations are operator-valued equations on
HS⊗ Φ of the form
dUt=
i,j=0
with initial condition U0 = I. The above equation has to be understood as an
integral equation
?t
i,j=0
the operators Li
jbeing bounded operators on HS alone which are ampliated to
HS⊗ Φ.
The main motivation and application of that kind of equation is that it gives
an account of the interaction of the small system HSwith the bath Φ in terms of
quantum noise perturbation of a Schr¨ odinger-like equation. Indeed, the first term
of the equation
dUt= L0
N
?
Li
jUtdai
j(t),
Ut= I +
0
N
?
Li
jUtdai
j(t),
0Utdt + ...
15
Page 16
describes the induced dynamics on the small system, all the other terms are quan-
tum noises terms. One of the main application of these equations is that they give
explicit constructions of unitary dilations of semigroups of completely positive
maps of B(HS) (see [H-P] and also section VII of this article).
Let us here only recall one of the main existence, uniqueness and boundedness
theorem connected to quantum Langevin equations. The literature is huge about
those equations; we refer to [Par] for the result we mention here. In the following,
by coherent vectors we mean elements of the space E generated by the u ⊗ ε(f),
with u ∈ HS, f ∈ L2(I R+;Cn) and
[ε(f)](σ) =
?
(s,i)∈σ
fi(s),
the usual coherent vectors of the Fock space Φ.
Theorem 5.–If all the operators Li
tic differential equation
jare bounded on HSthen the quantum stochas-
Ut= I +
N
?
i,j=0
?t
0
Li
jUsdai
j(s)
admits a unique solution defined on the space of coherent vectors.
The solution (Ut)t≥0is made of unitary operators if and only if there exist
on HS, a self-adjoint operator H, operators Li, i = 1,...,N and operators Si
i,j = 1,...,N such that the matrix (Si
j)i,j=1,...,N is unitary and the coefficients
Li
jare of the form
j,
L0
0= −(iH +1
2
N
?
k=1
L∗
kLk)
L0
j= Lj
Li
0= −
N
?
j− δijI.
k=1
L∗
kSk
i
Li
j= Si
III.5 Convergence theorems
We are finally able to state the main result of [AP1] which shows the conver-
gence of repeated interactions models to quantum stochastic differential equations.
Let τ be a parameter in I R+, which is thought of as representing a small time
interval. Let U(τ) be a unitary operator on HS⊗CN+1, with coefficients Ui
a matrix of operators on HS(this operator has to be though of as corresponding
to the unitary operator U of section II). Let Vk(τ) be the associated repeated
interaction operator:
Vk+1(τ) = Uk+1(τ)Vk(τ)
j(τ) as
16
Page 17
with the same notation as in section II. In the following we will drop dependency
in τ and write simply U, or Vk. Besides, we denote
εij=1
2(δ0i+ δ0j)
for all i,j in {0,...,N}. That is, for i,j ≥ 1
εi0= ε0j=1
2, εij= 0, ε00= 1.
Note that from now on we take the embeding of TΦ in Φ for granted and we
consider, without mentionning it, all the repeated quantum interactions to happen
in TΦ(τ), the subspace of Φ associated to the partition S = {ti= iτ;i ∈ I N}. The
main result of [AP1] (Theorem 13 in this reference) is the following.
Theorem 6.–Assume that there exist bounded operators Li
HSsuch that
lim
τ→0
for all i,j = 0,...,n. Then, for almost all t the operators V[t/τ]converge strongly,
when τ → 0, to Vt, the unitary solution of the quantum stochastic differential
equation
n
?
with initial condition V0= I.
j, i,j ∈ {0,...,n} on
Ui
j(τ) − δijI
τεij
= Li
j
dVt=
i,j=0
Li
jVtdai
j(t)
IV. The G.N.S. representation of the heat bath
In order to apply Theorem 6 to our repeated interaction model, we need the
state of the photon to be a vector state (i.e. a pure state) instead of a density
matrix. This is easily performed by considering the so-called G.N.S. representation
(or cyclic representation) of the photon system.
This representation can be described in the following way. Consider the space
H = L(CN+1) of endomorphisms of CN+1. Consider a given density matrix ρβ
on H, which is supposed to be in diagonal form ρβ= diag(β0,...,βN), where all
the βiare strictly positive. The space H = L(CN+1) is made into a Hilbert space
when equipped with the scalar product :
?A,B? = tr(ρβA∗B),
for all A,B ∈ H. The associated norm on H is denoted by ||·||. This Hilbert
space is (N +1)2-dimensional and we shall describe one of its orthonormal basis as
follows. We denote by X0
0the identity endomorphism. Then, for i = 1,...,N, we
put Xi
that
?Xi
ito be the diagonal matrices with diagonal coefficients {λ1
i,...,λN
i} such
i,Xj
j? = δij
17
Page 18
for all i,j = 0,1,...N. Such a family clearly exists for its diagonal elements are
obtained by extending the vector (1,...,1) ∈ CN+1into an orthonormal basis of
CN+1equiped with the scalar product
N
?
i=0
βixiyi.
For i ?= j ∈ {0,...,N} we put Xi
jto be the element of H given by
Xi
j=
√βi
1
ai
j.
It is then a straightforward computation to check that {Xi
an orthonormal basis of H.
We now have the usual G.N.S. representation π of L(CN+1) into L(H) given
by
π(A)B = AB,
for all A ∈ L(CN+1), B ∈ H. In the framework of that representation, note that
X0
for all A ∈ L(Cn+1) we have
?X0
That is, in the orthonormal basis we have choosen, X0
vector that could not be choosen to be different. The rest of the choice for our
orthonormal basis is just convenient for the computations, but the final result does
not depend on it.
j;i,j = 0,...N} forms
0is then the vector state on H which represents the state ρβon CN+1: indeed,
0,π(A)X0
0? = tr(ρβA).
0is the only important
Now, any operator K on HS⊗CN+1is transformed by π into an operator on
HS⊗ H. That is, π(K) is a (N + 1)2× (N + 1)2-matrix with coefficients Ki,j
L(HS). These coefficients are given by
Ki,j
k,lin
k,l= trH(ρβ(Xk
l)∗KXi
j)
where trH(H) denotes the partial trace of H along H, that is, this is the operator
on HS given by the sum of the diagonal coefficients of H as a L(HS)-valued
(N + 1)2× (N + 1)2-matrix.
When taking the continuous limit on the space HS⊗ ⊗I N∗H, we end up into
the space
HS⊗
?
I R+
H,
that is,
HS⊗ Γs
?
L2(I R+;C(N+1)2−1)
?
.
The associated quantum noises, following the same basis, are thus denoted by
dai,j
k,l(t), i,j,k,l = 0,...,N.
18
Page 19
V. The limit quantum Langevin equation
We are now in conditions to apply Theorem 6. We consider the repeated
interaction model described in section II, with its associated operators H, Uk, Vk.
Taking the G.N.S. representation of all that we end up in the space HS⊗⊗I N∗H,
which we embbed inside a continuous tensor product HS⊗ ⊗I R+H, as explained
in section III.3. The main result of this article is then the following.
Theorem 7.–In the continuous limit τ → 0, the repeated interaction dynamics
V[t/τ]converges strongly on HS⊗ Φ, for all t, to the (unitary) solution of the
quantum Langevin equation
?
i=0
2
?
+
βiV∗
dUt= −
iHS+ i
N
?
βiViUtdai,0
βiγiI +1
N
?
i=1
(β0V∗
iVi+ βiViV∗
i)
?
Utdt
− i
N
i=1
??
iUtda0,0
0,0(t) +
?
β0V∗
iUtda0,i
0,0(t)
?
i,0(t) +
?
β0ViUtda0,0
0,i(t)
?
. (8)
Proof
First, we represent the operators H and U on HS⊗ CN+1as (N + 1) ×
(N + 1)-matrices with coefficients in L(HS), following the orthonormal basis
{e0,e1,...,eN}. We get
1
√τVN
0
H(τ) =
HS+ γ0I
1
√τV∗
1
1
√τV∗
2
...
1
√τV∗
N
1
√τV1
HS+ γ1I0 ...0
1
√τV2
...
0
...
HS+ γ2I ...
...
...
0
...
···
0HS+ γNI
.
We need now to compute the associated unitary operator U = e−iτHin the
same framework. But as we wish to apply Theorem 6, we do not need to wonder
about the exact expression of U, but only the expansion of its coefficients in powers
of τ up to the pertinent orders given by Theorem 6. We get that U is represented
by the matrix
19
Page 20
I − iτHS− iτγ0I
i=1V∗
−i√τ V1+ o(τ3/2)
−i√τ V∗
1+ o(τ3/2) ...
−i√τ V∗
N+ o(τ3/2)
−1
2τ?N
iVi+ o(τ2)
I − iτHS− iτγ1I
−1
...
...
−1
2V1V∗
N+ o(τ2)
2τV1V∗
1+ o(τ2)
...
...
...
−i√τ VN+ o(τ3/2)
−1
2τVNV∗
1+ o(τ2) ...I − iτHS− iτγNI
−1
2VNV∗
N+ o(τ2)
.
This is for the expression of U as an operator on HS⊗ CN+1. Now our aim
is to compute the coefficients of the matrix of π(U) in the G.N.S. representation,
that is, as a (N + 1)2× (N + 1)2-matrix with coefficients in L(HS).
As we already discussed in section IV, the coefficients of π(U) are obtained
by computing the quantities
Ui,j
In order to get U0,0
0,0we have to compute
trH(ρβX0
This gives, using?N
U0,0
0,0= β0
2τ
?
?
N
?
We now compute the Ui,j
0,0terms, for i ?= j. That is, we compute
Ui,j
This trace is equal to
1
√βi
Then two distinct cases appear. If j = 0 we have
− i√τ VN+ o(τ3/2)
k,l= ?Xk
l,π(U)Xi
j? = trH(ρβ(Xk
l)∗UXi
j).
0UX0
0) = trH(ρβU).
i=0βi= 1
?
I − iτHS− iτγ0I −1
N
?
2τV1V∗
i=1
V∗
iVi+ o(τ2)
?
+
+ β1
I − iτHS− iτγ1I −1
I − iτHS− iτγNI −1
1+ o(τ2)
?
+ ...
?
+ βN
2τVNV∗
N+ o(τ2)
U0,0
0,0= I − iτHS− iτ
i=1
βiγiI −1
2τ
N
?
i=0
(β0V∗
iVi+ βiViV∗
i) + o(τ2). (9)
0,0= trH(ρβUXi
j).
?ei,ρβUej? =
?
2τ?N
...
βi?ei,Uej?.
Ue0=
I − iτHS− iτγ0I −1
i=1V∗
iVi+ o(τ2)
− i√τ V1+ o(τ3/2)
20
Page 21
and thus
Ui,0
0,0= −i√τ
?
βiVi+ o(τ3/2). (10)
But when j ?= 0 we have
Uej=
−i√τ V∗
−1
j+ o(τ3/2)
2τV1V∗
j+ o(τ2)
...
2τVjV∗
...
j+ o(τ2)
I − iτHS− iτγjI −1
j+ o(τ2)
−1
?
?
2τVNV∗
.
Now, if i = 0 we get
U0,j
0,0= −i
β0
√τ V∗
j+ o(τ3/2), (11)
if i ?= 0 and i ?= j we get
Ui,j
0,0= −1
2
βiτ ViV∗
j+ o(τ2). (12)
A similar computation gives
U0,0
0,j= −i√τ
U0,0
i,0= −i
?
βi
β0Vj+ o(τ3/2),
√τ V∗
(13)
?
?
i+ o(τ3/2) (14)
and, still for i ?= j and i,j ?= 0
U0,0
i,j= −1
2
βiτ VjV∗
i+ o(τ2). (15)
Now, consider i ?= j and k ?= l, we have
Ui,j
k,l= trH(ρβ(Xk
l)∗UXi
j) =
?
jep?
p
?Xk
lρβep,UXi
jep?
=
?
p
βp
1
√βkβi?ak
lep,Uai
= δik?el,Uej?.
That is, if l = 0 and j ?= 0
Ui,j
k,0= δik(−i√τV∗
j+ o(τ3/2))(16)
and for l = j = 0
Ui,0
k,0= δik(I − iτHS− iτγ0I −1
2
N
?
i=1
V∗
iVi+ o(τ2)). (17)
If l ?= 0 and l ?= j with j ?= 0 we have
Ui,j
k,l= δik(−1
2τ VlV∗
j+ o(τ2)), (18)
if l = j ?= 0
Ui,j
k,j= δik(I − iτHS− iτγlI −1
2τ VlV∗
l+ o(τ2))(19)
21
Page 22
and finally if l ?= 0 and j = 0 then
Ui,0
k,l= δik(−i√τVl+ o(τ3/2)).
j,jterms. Recall that the Xi
i,...λN
with the scalar product?
Ui,i
(20)
We now study the Ui,i
coefficients (λ0
iare diagonal matrices whose
i), i = 0,...,N form an orthonormal basis of CN+1equipped
iβixiyi. We get
0,0= tr(ρβUXi
i) =
N
?
k=0
?ek,ρβUXi
iek?
=
N
?
N
?
k=0
βkλk
i?ek,Uek?
=
k=1
βkλk
i(I − iτHS− iτγkI −1
2τ VkV∗
k+ o(τ2))
+ β0λ0
i(I − iτHS− iτγ0I −1
2
N
?
i=1
V∗
iVi+ o(τ2))
=
N
?
k=1
βkλk
i(−iτγkI −1
2τ VkV∗
k+ o(τ2))
+ β0λ0
i(−iτγ0I −1
2
N
?
i=1
V∗
iVi+ o(τ2))(21)
where we have used?
kβkλk
i= 0. In the same way,
U0,0
i,i=
N
?
k=1
βkλk
i(−iτγkI −1
2τ VkV∗
k+ o(τ2))
+ β0λ0
i(−iτγ0I −1
2
N
?
i=1
V∗
iVi+ o(τ2)). (22)
Finally, using?N
Ui,i
k=0βkλk
jλk
i= δij, we obtain
j,j=
N
?
k=0
βkλk
jλk
i?ek,Uek?
= δij(I − iτHS) +
N
?
k=1
βkλk
jλk
i(−iτγkI −1
2τ VkV∗
k+ o(τ2))
+ β0λ0
jλ0
i(−iτγ0−1
2τ
N
?
j=1
V∗
jVj+ o(τ2)). (23)
The last terms to be considered are those of type Ui,i
k,l(and conversely Uk,l
i,i), with
22
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k ?= l and i ?= 0. We have
Ui,i
k,l= tr(ρβ(Xk
l)∗UXi
i) =
?
p
?ep,ρβ(Xk
l)∗UXi
iep?
=
?
p
βpλp
i?Xk
lep,Uep?
= βkλk
i
1
√βk?el,Uek?
which gives
?
?
β0λ0
i(−i√τVl+ o(τ3/2))(24)
or
βkλk
i(−1
2τVlV∗
k+ o(τ2))(25)
depending on k = 0 or not. The case of Uk,l
for anyway it will not contribute to the continuous limit.
i,iis similar and needs not be explicited
We can now apply Theorem 6. Following the rules of Theorem 6, we need to
check that there exists bounded operators Li,j
k,lon HSsuch that
s − lim
τ→0
Ui,j
k,l− δ(i,j),(k,l)I
τεi,j
k,l
= Li,j
k,l,
where ε0,0
Equality (9) shows that
0,0= 1, ε0,0
k,l= εk,l
0,0= 1/2 and the others εi,j
k,lare equal to 0.
L0,0
0,0= −iHS− i
N
?
i=1
βiγiI −1
2
?
i
(β0V∗
iVi+ βiViV∗
i).
By (10) we have
Li,0
0,0= −i
?
?
βiV∗
?
βiVi
and in the same way
L0,i
0,0= −i
L0,0
i,0− i
L0,0
0,i= −i
β0V∗
i
?
i
β0Vi
by (11), (14) and (13) respectively.
The other terms Ui,j
only their√τ part contributes to the limit. As a consequence Li,j
Terms of the form Ui,j
k,l(equations (16) to (20)) contribute in the limit via the
order 1 terms in Ui,j
k,l− δ(i,j),(k,l)I, that is 0 in anycase (the I term in (17) and
(19) does not contribute as it indeed appears only when (i,j) = (k,l).
The same holds for Ui,i
and (22)).
The terms Ui,i
Following (23) we get a null contribution in all cases.
0,0and U0,0
i,jappear to be of order τ in (12) and (15), while
0,0= L0,0
i,j= 0.
0,0and U0,0
i,iwhich gives Li,i
0,0= L0,0
i,i= 0 (equations (21)
j,jcontribute in the limit via the order 1 terms of Ui,i
j,j− δijI.
23
Page 24
Finally, equality (24) and (25) show that the last coefficients also vanish in
the limit.
This exactly gives the announced quantum Langevin equation.
Remark: One can only be impressed (at least that was the case of the
authors when performing the computations) by the kind of “mathematical miracle”
occuring here: all the (N + 1)2terms fit perfectly in the type of conditions of
Theorem 6. The number of cancellation one may hope for happens exactly.
Equation (8) takes a much more useful form if one regroups correctly the
different terms. Indeed, put
?
β0− βi
?
and
Wi= −i
then the equation (8) simply writes
?
i=0
2
i=1
?
A0
i(t) =
β0
a0,0
0,i(t) +
?
?
βi
β0− βi
βi
β0− βia0,0
ai,0
0,0(t) (26)
Ai
0(t) =
β0
β0− βia0,i
0,0(t) +
i,0(t) (27)
?
?
β0− βiVi
dUt= −
iHS+ i
N
?
βiγiI +1
N
?
β0
β0− βiW∗
0(t)?.
iWi+
βi
β0− βiWiW∗
i
??
Utdt
+
i
?WiUtdA0
i(t) − W∗
iUtdAi
(28)
VI. Thermal quantum noises and their properties
In this section we concentrate on the particular quantum noises (26) and (27)
that appeared above. We shall show that they are natural candidates for being
qualified as “thermal quantum noises”. We also show that the form of equation
(25) is the generic one for unitary solutions, in the thermal case.
The situation that has appeared in the previous section can be summarized
as follows.
We consider the quantities β, γiand thus βias being fixed.
First of all, there is no need to consider a Fock space over L2(I R+;C(N+1)2−1)
anymore, for most of the quantum noises ai,j
(8). More economical is to consider a double Fock space:
?Φ = Γs(L2(I R+,CN)) ⊗ Γs(L2(I R+,CN)).
ai
k,l(t) do not play any role in equation
Each of the copies of the Fock space accomodates the quantum noises
j(t) ⊗ I and I ⊗ ai
respectively, which we shall denote more simply by
ai
j(t)
j(t) and bi
j(t)
24
Page 25
respectively.
Form the operator processes
A0
i(t) =
?
?
β0
β0− βia0
β0
β0− βiai
i(t) +
?
?
βi
β0− βibi
βi
β0− βib0
0(t) (28)
Ai
0(t) =
0(t) +
i(t). (29)
For every f ∈ L2(I R+;Cn) with coordinates (fi) in the basis {e1,...,en} put
A∗(f) =
n
?
i=1
?
I R+fi(t)dA0
i(t)
and
A(f) =
n
?
i=1
?
I R+fi(t)dAi
0(t).
Proposition 8.–The operators A(f),A∗(g) form a non-Fock representation of
the CCR algebra over (L2(I R+;CN)).
Proof
The operators A(f) and A∗(g) have similar properties as the usual quantum
noises. In particular, they admit a quantum stochastic integration theory, which
is completely identical to the usual one. This does not need to be developed here.
We shall only prove that the quantum Ito formula (7) is now driven by the rules:
dAi
0(t)dA0
i(t) =
β0
β0− βidt
βi
β0− βi
and
dA0
i(t)dAi
0(t) =
dt.
Indeed, we have
dAi
??
β0− βi
β0
β0− βi
and
dA0
??
β0− βi
βi
β0− βi
0(t)dA0
i(t) =
=
β0
dai
0(t) +
?
βi
β0− βi
db0
i(t)
???
β0
β0− βi
da0
i(t) +
?
βi
β0− βi
dbi
0(t)
?
= da0
0(t) =
β0
β0− βi
dt
i(t)dAi
0(t) =
=
β0
da0
i(t) +
?
βi
β0− βi
dbi
0(t)
???
β0
β0− βi
dai
0(t) +
?
βi
β0− βi
db0
i(t)
?
= db0
0(t) =
βi
β0− βi
dt.
25
Page 26
By the quantum Ito formula we get
[A(f),A∗(g)] =
N
?
i=1
?
I R+fi(t)gi(t)
?β0− βi
β0− βi
?
dtI
= ?f,g?I.
In other words, the operators A(f),A∗(g) form a representation of the CCR al-
gebra over (L2(I R+;CN)). But this clearly a non-Fock one for the creation and
annihilation operator attached to this representation do not generate the whole
creation and annihilation operators of the underlying (double) Fock space.
Now, let us form the associated Weyl operators
W(f) = exp
?A(f) + A∗(f)
√2
?
.
We wish to compute the statistics of W(f) in the vaccum state Ω. For this purpose,
we use the following notation. If H is any operator on CN+1, then it acts on
L2(I R+;CN) by
[Hf](s) = λ0+
N
?
i=1
λifi(s).
This has to be understood as follows: in general λ0 is chosen to be equal to 0,
thus H acts on L2(I R+;CN) as a multiplication operator. In our case it is the
multiplication by a constant (vector).
Theorem 9.–We have
?Ω,W(f)Ω? = exp
?
−1
4?f,coth(βHR
2
)f?
?
for all f ∈ L2(I R+;CN).
Proof
We have
A(f) + A∗(f) =
N
?
i=1
?∞
0
??
β0
β0− βifi(s)da0
?
i(s) +
?
βi
β0− βifi(s)dbi
?
0(s) +
+
β0
β0− βifi(s)dai
0(s) +
βi
β0− βifi(s)db0
i(s)
?
.
If we put
a(f) =
N
?
i=1
?∞
0
?
fi(s)da0
i(s) + fi(s)dai
0(s)
?
and
b(f) =
N
?
i=1
?∞
0
?
fi(s)db0
i(s) + fi(s)dbi
0(s)
?
26
Page 27
then the above expression shows that
A(f) + A∗(f) = a(?f) + b(?f)
?
β0− βi
?
where
?fi(s) =
?fi(s) =
β0
fi(s)
βi
β0− βifi(s).
Denote by Waand Wbthe usual Weyl operators associated to the noises a and b
respectively. We have clearly shown that
W(f) = Wa(?f) ⊗ Wb(?f).
?
?
4
i=1
?
4
i=1
?
As a consequence, using usual computations on the Weyl operators
?Ω,W(f)Ω? = exp
−1
4
????
????f
??????
2
+
??? ????f
??????
+
2??
= exp
−1
n
?
n
?
?
?
β0
β0− βi
coth(βγi
βi
β0− βi
?
||fi||2
?
= exp
−1
2)||fi||2??
?
= exp
−1
4?f,coth(βHR
2
)f?
.
We recover an analogue of the usual K.M.S. state statistics for a free Boson
gas at thermal equilibrium. Let us discuss that point more precisely. Usually,
the Hamiltonian model for a quantum heat bath is as follows. We are given a
function ω(s) (in Fourier representation actually, by this does not matter much
here) and the Hamiltonian of the heat bath, on the Fock space Γs(L2(I R+;C)) is
the differential second quantization operator dΓ(ω) associated to the multiplication
by ω.
In our discrete model, if we take the typical interacting system HRto be C2
but with a Hamiltonian depending on the number of the copy:
?0
then in the continuous limit, the corresponding Hamiltonian on the Fock space
Γs(L2(I R+;C)) is indeed dΓ(ω) (under some continuity assumption on ω, cf [AP2]).
In the case we have described here, the situation is made a little more com-
plicated by the fact that we considered a chain of CN+1instead of C2, but a lot
easier by taking a constant Hamiltonian HR. The time-dependent case stays to
be explored, no doubt it will give rise to the usual free Bose gaz statistics.
HR(k) =
0
0ω(k)
?
27
Page 28
Remark : The parameter β used here is supposed to be the inverse of the
temperature of the heat bath (more exactly 1/kT). If we make the temperature
go to 0, that is, β goes to +∞, then
β0
β0− βi
converges to 1, for γi− γ0> 0 by hypothesis, and
βi
β0− βi
converges to 0. This makes all the noises bi
g. We recover the usual quantum noises, the usual Weyl operators. This means
that the usual quantum noises are the 0 temperature ones.
=
1
1 − e−β(γi−γ0)
=
e−βγi
1 − e−β(γi−γ0)
jbeing useless and A(g) = a(g), for all
We now describe which kind of quantum Langevin equation, driven by those
thermal quantum noises gives rise to a unitary evolution.
Theorem 10.–A Langevin equation of the form
dUt= K0
0Utdt +
n
?
i=0
?K0
iUtdA0
i(t) + Ki
0UtdAi
0(t)?,U0= I,
where the coefficients Ki
solution on the set of coherent vectors. The solution is unitary if and only if it is
of the form
?
2
i=1
jare all bounded operators on HS, always admit a unique
dUt=
−iH −1
n
?
?
β0
β0− βiW∗
iWi+
βi
β0− βiWiW∗
i
??
Utdt+
+
n
?
i=0
?WiUtdA0
i(t) − W∗
iUtdAi
0(t)?
for some bounded operators Wi, i = 1,...,n, on HS and a self-adjoint bounded
operator H on HS.
Proof
The existence and uniqueness result is a simple consequence of the one quoted
in Theorem 5.
For characterizing the unitarity we use algebraical (formal) computations,
which are the same as for the proof of Theorem 5. The analytical part of the proof
is totaly identical to the one of Theorem 5. There is no need to develop it here.
Our equation is of the general form
n
?
We thus also have
n
?
dUt= K0
0Utdt +
i=0
?K0
iUtdA0
i(t) + Ki
0UtdAi
0(t)?
dU∗
t= U∗
t(K0
0)∗dt +
i=0
?U∗
t(K0
i)∗dAi
0(t) + U∗
t(Ki
0)∗dA0
i(t)?.
28
Page 29
By the quantum Ito formula we get
d(U∗
tUt) = (dU∗
t)Ut+ U∗
tdUt+ dU∗
n
?
n
?
U∗
tdUt
= U∗
t(K0
0)∗Utdt +
i=0
?U∗
?U∗
i)∗K0
t(K0
i)∗UtdAi
0(t) + U∗
t(Ki
0)∗UtdA0
i(t)?+
+ U∗
tK0
0Utdt +
i=0
tK0
iUtdA0
i(t) + U∗
tKi
0UtdAi
0(t)?+
t(Ki
+
β0
β0− βi
?
n
?
n
?
i=1
t(K0
iUtdt +
βi
β0− βi
n
?
i=1
U∗
0)∗Ki
0Utdt
= U∗
t
(K0
0)∗+ K0
0+
n
?
i=1
?
?Ut
β0
β0− βi(K0
?dAi
i)∗K0
i+
βi
β0− βi(Ki
?U∗
0)∗Ki
0
??
?dA0
Utdt
+
i=0
?U∗
t
?(K0
i)∗+ Ki
00(t) +
n
?
i=0
t
?(Ki
0)∗+ K0
i
?Ut
i(t).
By a similar computation we obtain
d(UtU∗
?
t) =
=UtU∗
t(K0
0)∗+ K0
0UtU∗
t+
n
?
i=1
?
β0
β0− βiKi
0UtU∗
t(Ki
0)∗+
+
βi
β0− βiK0
?UtU∗
iUtU∗
t(K0
i)∗
??
dt
+
n
?
i=0
?UtU∗
t(K0
i)∗+ Ki
0UtU∗
t
?dAi
0(t) + +
n
?
i=0
t(Ki
0)∗+ K0
iUtU∗
t
?dA0
i(t).
Asking both to be equal to 0 for every t is equivalent to the following conditions :
Ki
n
?
Put
0+1
2
i=1
the last condition above exactly says
K = −K∗.
We thus obtain the announced characterization.
0= −(K0
(K0
i)∗
0)∗+ K0
0+
i=1
?
β0
β0− βi(K0
i)∗K0
i+
βi
β0− βiK0
i(K0
i)∗
?
= 0.
K = K0
n
?
?
β0
β0− βi(K0
i)∗K0
i+
βi
β0− βiK0
i(K0
i)∗
?
,
The attentive reader has noticed that this is exactly the form of equation (28)!
VII. The Lindblad generator
Going back to the usual Langevin equations of Theorem 5, let us recall a very
important theorem, which is the main point in using quantum Langevin equations
in order to dilate quantum dynamical semigroups.
29
Page 30
Theorem 11.–Consider the unitary solution (Ut)t≥0of the quantum Langevin
equation
dUt= −(iH +1
2
N
?
k=1
L∗
kLk)Utdt +
N
?
i=0
WiUtda0
i(t) −
n
?
i=0
W∗
kUtdai
0(t).
For any bounded operator X on HS, the application
t ?→ Pt(X) = ?Ω,U∗
is a semigroup of completely positive maps whose Lindblad generator is
t(X ⊗ I)UtΩ?
L(X) = i[H,X]−1
2
N
?
i=1
(W∗
iWiX + XW∗
iWi− 2W∗
iXWi).
In our thermal case we have the following the form for the Lindblad generator.
Theorem 12.–Consider the unitary solution (Ut)t≥0of the thermal quantum
Langevin equation
?
2
i=1
dUt=
−iH −1
N
?
?
β0
β0− βiW∗
iWi+
βi
β0− βiWiW∗
?
i
??
Utdt+
+
N
i=0
?WiUtdA0
i(t) − W∗
iUtdAi
0(t)?.
For any bounded operator X on HS, the application
t ?→ Pt(X) = ?Ω,U∗
is a semigroup of completely positive maps whose Lindblad generator is
t(X ⊗ I)UtΩ?
L(X) = i[H,X]−1
2
N
?
N
?
i=1
β0
β0− βi(W∗
βi
β0− βi
iWiX + XW∗
iWi− 2W∗
iXWi)
−1
2
i=1
(WiW∗
iX + XWiW∗
i− 2WiXW∗
i).
Proof
The basic computation is the same as for Theorem 11:
– By the “thermal quantum Ito formula” (see proof of Proposition 8) one
computes
d(U∗
t(X ⊗ I)Ut) = (dU∗
– The only contributing term when averaging over the vacuum state is the
coefficient of dt, that is, we get the equation
d?Ω,U∗
The solution is clearly a semigroup with generator L.
t)(X ⊗ I)Ut+ U∗
t(X ⊗ I)(dUt) + (dU∗
t)(X ⊗ I)(dUt);
t(X ⊗ I)UtΩ? = ?Ω,U∗
t(L(X) ⊗ I)UtΩ?dt.
30
Page 31
Let us write down in a corollary, the Lindblad generator in perspective with
the initial Hamiltonian.
Corollary 13–If the repeated interaction model is having the following total
Hamiltonian:
H = HS⊗ I + I ⊗ HR+
1
√τ
N
?
i=1
?Vi⊗ a0
i+ V∗
i⊗ ai
0
?
then the associated Lindblad generator in the continuous limit is
L(X) = i[HS,X] −1
2
N
?
N
?
i=1
β0(V∗
iViX + XV∗
iVi− 2V∗
iXVi)
−1
2
i=1
βi(ViV∗
iX + XViV∗
i− 2ViXV∗
i).
VIII. Thermalization
In this section we answer a very natural question in this context. Consider
a given quantum system HS with a given Hamiltonian HS. Is there a natural
Lindblad generator L (in the Schr¨ odinger picture) on HSwhich admits as a unique
invariant state, the state
1
Zβ
and which possesses the property of return to equilibrium for this state? By “return
to equilibrium” we mean the following: for every inital state ρ0, the evolution
etL(ρ0) converges to the state ρβin the ∗-weak sense, that is,
lim
ρβ=
e−βHS
t→+∞tr(etL(ρ0)X) = tr(ρβX)
for all observable X.
We shall prove in this section that the answer to the above question is positive,
at least if HSis finite-dimensional. For this purpose we recall a famous result by
Frigerio and Veri [F-V], in a slightly extended form due to Fagnola and Rebolledo
[F-R].
Theorem 14–Let
L(ρ) = −i[H,ρ] −1
2
n
?
i=1
(L∗
iLiρ + ρL∗
iLi− 2LiρL∗
i)
be a Lindblad generator (in Schr¨ odinger picture). If the commutants
{H,Li,L∗
coincide then the associated dynamics possesses the property of return to equilib-
rium.
i;i = 1,...n}′
and
{Li,L∗
i;i = 1,...n}′
31
Page 32
We consider HS a N + 1-dimensional Hilbert space, with Hamiltonian (in
diagonal form)
0 ...
Consider the Gibbs state ρβ = (1/Zβ)e−βHS, it is also of diagonal form with
diagonal elements denoted by β0,β1,...,βN.
We put the system HSin repeated quantum interaction with a chain of copies
of CN+1with the total Hamiltonian
HS=
λ0
0
...
0 ...
...
...
...
0
0
...
λ1
λN
.
H = HS⊗ I + I ⊗ HR+
1
√τ
N
?
i=1
?Vi⊗ a0
i+ V∗
i⊗ ai
0
?,
where Viis the matrix
Vi=
0
0
...
0
...
0
1 ...0
0
...
0
...
0 ...
with the 1 being at the i-th row and where HR= HS.
Note that this means that we put the system HRin repeated quantum inter-
action with a chain of copies of ... itself but in the desired state. This is somehow
very natural!
Theorem 15–In the continuous limit the above repeated interaction model admits
the following Lindblab generator in the Schr¨ odinger picture:
L(ρ) = −i[HS,ρ] −1
2
N
?
N
?
i=1
β0(V∗
iViρ + ρV∗
iVi− 2ViρV∗
i)
−1
2
i=1
βi(ViV∗
iρ + ρViV∗
i− 2V∗
iρVi).
This Lindblad generator admits
ρβ=
1
Zβ
e−βHS
as a unique invariant state and it converges to equilibrium.
Proof
The announced Lindblad generator is just the dual of the Lindblad generator
described in Corollary 13.
Let us compute L(ρβ). We have
[HS,ρβ] = 0.
32
Page 33
On the other hand, the Vi’s have been chosen so that
ρβVi=βi
β0Viρβ.
This gives
V∗
iρβ=βi
β0ρbV∗
i
These two relation give
V∗
iViρβ+ ρβV∗
iVi− 2ViρV∗
i= 2V∗
iViρβ− 2β0
βiViV∗
iρβ
and
ViV∗
iρβ+ ρβViV∗
i− 2V∗
iρVi= 2ViV∗
iρβ− 2βi
β0V∗
iViρβ.
Hence we get the result: L(ρβ) = 0.
Let us consider the von Neumann algebra generated by the operators Vi, V∗
i = 1...N. It is easy to see that this is the whole B(HS). Hence the commutant
of this von Neumann algebra is trivial. As we always have the obvious inclusion
{HR,Vi,V∗
we have equality of the two commutants and Theorem 14 applies. This gives the
return to equilibrium property and hence the uniqueness of the invariant state.
i,
i;i = 1,...n}′⊂ {Vi,V∗
i;i = 1,...n}′
Note the important following fact: we never used the fact that ρβis a Gibbs
state, we only used the fact that it is a function of HR. Hence the above result is
valid for any state ρ which is a function of HR.
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[A-W] H. Araki, E.J. Woods: “Representation of the canonical commutation rela-
tions describing a nonrelativistic infinite free Bose gas”, Journal of Mathematical
Physics 4 (1963), p. 637-662.
[At1] S. Attal: “Extensions of the quantum stochastic calculus”, Quantum Prob-
ability Communications vol. XI, World Scientific (2003), p. 1-38.
[At2] S. Attal: “Quantum Noise Theory”, book to appear, Springer Verlag.
[AJ1] S. Attal, A. Joye: “Weak-coupling and continuous limits for repeated quan-
tum interactions”, preprint.
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